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Question Regarding The Odds of The Pennsylvania Millionaire Raffle

Topic closed. 23 replies. Last post 11 months ago by LottoMetro.

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LottoMetro's avatar - Lottery-024.jpg
Happyland
United States
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September 1, 2013
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Posted: January 16, 2016, 2:35 pm - IP Logged

Anytime the lottery uses the terms "odds" or "chances" they are illustrating probability based on combinatorics. Their usage is not the same as it is in casino games or horse racing. The difference is purely semantic and though academics frequently point this out when reviewing lottery terminology, nobody really cares.

If you take a hypergeometric distribution with sample success of 1, population success of 4, sample size of 1 and population size of 500k, you will calculate that the probability of winning one of the top prizes is 0.0008%. Divide 1 by this and you will get 125,000. The only way this probability changes is if fewer than 500k tickets are sold, and then you would simply plug in the number of tickets sold as the population size to determine the true odds. I use this methodology for analyzing scratch-off games. The probabilities/chances/odds posted by the lottery are mathematically correct but assume all 500k tickets are sold.

Also, if there is no mathematical difference between what has been stated in above posts and 1:125000, why would the lottery confuse the hell out of players by posting anything other than the simplified odds? It is impractical for them to advertise "live" odds based on sales. Again, the debate is purely semantics.

If the chances of winning the jackpot are so slim, why play when the jackpot is so small? Your chances never change, but the potential payoff does.
If a crystal ball showed you the future of the rest of your life, and in that future you will never win a jackpot, would you still play?

2016: -48.28% (13 tickets) ||
P&L % = Total Win($)/Total Wager($) - 1

    JADELottery's avatar - MeAtWork 03.PNG
    The Quantum Master
    West Concord, MN
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    Posted: January 16, 2016, 4:37 pm - IP Logged

    well that's odd.

    we ran through a billion raffle samples and the lowest odds based on sample simulation we came up with in the top 4 prizes was around 1 : 484026.

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    Use at your own risk.

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    Jehocifer

      LottoMetro's avatar - Lottery-024.jpg
      Happyland
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      Posted: January 16, 2016, 8:19 pm - IP Logged

      well that's odd.

      we ran through a billion raffle samples and the lowest odds based on sample simulation we came up with in the top 4 prizes was around 1 : 484026.

      Not sure a simulation is useful unless you believe the sales follow a specific distribution and/or will be less than the maximum allowed tickets. According to PA lottery records, all past Millionaire Raffle drawings have sold out, which pretty much satisfies the advertised odds. As the simulation approaches infinity it will converge on the theoretical (advertised) probabilities.

      Just for kicks I ran a simulation of about 2.5 M trials using a beta distribution* with range of sales from 300k to 500k. Due to the shape of the distribution there was lower probability of both lower extreme (~300k) and higher extreme sales (~500k). Nevertheless, the minimum odds were approximately 1 in 75,715 and the maximum odds were approximately 1 in 124,989 (you can multiply each by 4 if the larger numbers are more appealing). All related to the tickets sold. This illustrates the beauty of the central limit theorem.

       

      *Beta is continuous; whereas in reality, ticket sales will be discrete. However, simulating on a discrete distribution would result in precisely the same probabilities as can be worked out using hypergeometric distribution and combinatorics. Basically making simulations pointless :-)

      If the chances of winning the jackpot are so slim, why play when the jackpot is so small? Your chances never change, but the potential payoff does.
      If a crystal ball showed you the future of the rest of your life, and in that future you will never win a jackpot, would you still play?

      2016: -48.28% (13 tickets) ||
      P&L % = Total Win($)/Total Wager($) - 1

        JADELottery's avatar - MeAtWork 03.PNG
        The Quantum Master
        West Concord, MN
        United States
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        Posted: January 17, 2016, 1:46 pm - IP Logged

        We will redirect to the PA Lottery's own Millionaire Raffle #23 rules - http://www.pabulletin.com/secure/data/vol46/46-2/56.html.

        "...

         10. Number and Description of Prizes and Approximate Odds: The Pennsylvania Millionaire Raffle XXIII lottery game prizes and determination of winners are as follows:

        Ticket Matching Exactly the Unique Eight-digit Number Drawn:Win Prize Of:Maximum Odds
        Of Winning Are 1 In:
        Number Of Winners
        First-Prize-Tier$1,000,000125,0004
        Second-Prize-Tier$100,000125,0004
        Third-Prize-Tier$1,0005,000100
        Fourth-Prize-Tier$10084.865,892

         

        The odds of winning are based on selling all 500,000 tickets. If all 500,000 tickets are not sold, the odds of winning will depend on the total number of tickets sold. All Pennsylvania Millionaire Raffle XXIII lottery game prize payments, including first-prize-tier prizes, will be made as one-time, lump-sum cash payments. Federal income tax withholding will be automatically deducted from the lump-sum cash payment for all first-prize-tier and second-prize-tier prizes.

        ..."

        The posted prize-odds matrix by the PA Lottery is based on All 500,000 Tickets sold and that's All we are addressing here.

        Had less than 500,000 tickets been sold, the above matrix becomes invalid.

        Presented 'AS IS' and for Entertainment Purposes Only.
        Any gain or loss is your responsibility.
        Use at your own risk.

        Order is a Subset of Chaos
        Knowledge is Beyond Belief
        Wisdom is Not Censored
        Douglas Paul Smallish
        Jehocifer

          GiveFive's avatar - Lottery-026.jpg
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          Posted: January 17, 2016, 2:42 pm - IP Logged

          There's been a lot of excellent discussion here regarding odds/chances/probability of the PA Millionaire raffle, and I thank everyone that has contributed to the discussion.

          That said, I would respectfully disagree that all Millionaire Raffles have sold out.  During the sales cycle of the most recent one, ~474,000 tickets were sold. (Someone here at The LP posted that info, which they said they obtained from the PA Lottery's website.) I believe too, that the Halloween Millionaire raffle held on October 31st 2014 didn't sell out.  But I'm quibbling over basically nothing.  Thanks again for the high quality discussion.  It's a lot more in depth than I had hoped for, and I'm certainly a better player for having read it.  G5 

          About playing the lottery --  You will lose more than you win. Until you hit a jackpot.  Then everything changes!

            sirbrad's avatar - Lottery-062.jpg
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            Posted: January 17, 2016, 2:46 pm - IP Logged

            They used to sell out all the time, but now they seem to have stopped. They used to advertise them when they go real low but have not seen that lately. So sometimes I get one for better chances yet. I won $100 last year on one ticket and was close to the $1,000 twice, this year by two numbers. They probably don't sell out as much now though as $20.00 is a lot to keep losing and 1 million is not all that much these days. Not to mention the controversial draw error this time, and no one knowing for sure if the tickets sold after 8PM were counted in the draw. So I do not play it as much. I would rather spend $30 on better games or 20 MM than that game. I only get one ticket now maybe 2-3 draws.

              sirbrad's avatar - Lottery-062.jpg
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              Posted: January 17, 2016, 2:55 pm - IP Logged

              Here's something else I'm curious about regarding the raffle.

              Can a player win a prize twice with the same ticket number?

              I would think that once ticket number,  "00123456" had won a prize, it would no longer be eligible to win another one, ie that it would be removed from the "prize pool" of 500,000 possible winners.  Anybody know?

              And... are the first 4 numbers "drawn" (and I'm using that term loosely) the top prize winners?  I guess there's a lot I don't know about the raffle!  G5

              Nope, much like a barrel or hat raffle one number wins one prize that is it. The biggest prizes are all drawn first.

              5. Prizes: The prizes that can be won in this game are $100, $1,000, $100,000 and $1,000,000. A player may only win one time on each ticket or chance.

                JADELottery's avatar - MeAtWork 03.PNG
                The Quantum Master
                West Concord, MN
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                Posted: January 17, 2016, 4:19 pm - IP Logged

                There's been a lot of excellent discussion here regarding odds/chances/probability of the PA Millionaire raffle, and I thank everyone that has contributed to the discussion.

                That said, I would respectfully disagree that all Millionaire Raffles have sold out.  During the sales cycle of the most recent one, ~474,000 tickets were sold. (Someone here at The LP posted that info, which they said they obtained from the PA Lottery's website.) I believe too, that the Halloween Millionaire raffle held on October 31st 2014 didn't sell out.  But I'm quibbling over basically nothing.  Thanks again for the high quality discussion.  It's a lot more in depth than I had hoped for, and I'm certainly a better player for having read it.  G5 

                Thank you.

                If we helped to your satisfaction, we'll consider this closed and no more replies.

                Good Luck.

                Presented 'AS IS' and for Entertainment Purposes Only.
                Any gain or loss is your responsibility.
                Use at your own risk.

                Order is a Subset of Chaos
                Knowledge is Beyond Belief
                Wisdom is Not Censored
                Douglas Paul Smallish
                Jehocifer

                  LottoMetro's avatar - Lottery-024.jpg
                  Happyland
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                  September 1, 2013
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                  Posted: January 17, 2016, 5:42 pm - IP Logged

                  There's been a lot of excellent discussion here regarding odds/chances/probability of the PA Millionaire raffle, and I thank everyone that has contributed to the discussion.

                  That said, I would respectfully disagree that all Millionaire Raffles have sold out.  During the sales cycle of the most recent one, ~474,000 tickets were sold. (Someone here at The LP posted that info, which they said they obtained from the PA Lottery's website.) I believe too, that the Halloween Millionaire raffle held on October 31st 2014 didn't sell out.  But I'm quibbling over basically nothing.  Thanks again for the high quality discussion.  It's a lot more in depth than I had hoped for, and I'm certainly a better player for having read it.  G5 

                  PA didn't have a Halloween raffle in 2014 but you're right in that not all have sold out (I believe up until they offered more than 1 per year they did all sell out). The one this past October only sold 356,904 tickets. The ones offered this year seem to have better "padding" built into the timeframe that tickets are made available.

                  As I stated earlier you can determine the probability of there being a winner based on tickets sold by simply using a hypergeometric distribution which unsurprisingly, is approximately equivalent to dividing the number of tickets sold by the number of prizes available.

                  For the drawing this past October that comes to chances of 1 in 89,226. Better than most $20 scratch-offs but the overall payout (roughly 71.3 cents per dollar) was worse than many scratchoffs in that price point. They would have to sell around 300K or fewer tickets to become a better proposition overall compared to high-tier scratchers.

                  If the chances of winning the jackpot are so slim, why play when the jackpot is so small? Your chances never change, but the potential payoff does.
                  If a crystal ball showed you the future of the rest of your life, and in that future you will never win a jackpot, would you still play?

                  2016: -48.28% (13 tickets) ||
                  P&L % = Total Win($)/Total Wager($) - 1